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Let A be a principally polarized abelian variety of dimension g over a number field K. Assume that the image of the adelic Galois representation of A is an open subgroup of GSp_{2g}(\hat {\mathbb {Z}}). Then there exists a positive integer m so that the Galois image of A is the full preimage of its reduction modulo m. The least m with this property, denoted m_A, is called the image conductor of A. Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for m_A, in terms of standard invariants of A, in the case that A is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension. [ABSTRACT FROM AUTHOR]